Ratios and Double Number Lines

Students work on their own on the Think About It problem.  This is a quick problem for students to complete.  Students fill in the ratio table.

As a class, we discuss why a tape diagram could not be the visual model for this problem.  I frame the lesson by telling students that, in this lesson, they will learn a third model that can be used with ratios.  Specifically, double number lines are used with part to part rations that have different units.

In the Intro to New Material section, students learn to create the double number line model.  I guide students to create the double number line for the first example problem.  We need to use two number lines because we're representing quantities with different units.  I tell students that it would not make sense to model minutes and miles on the same number line.

Using these steps, we fill in the notes in this section:

Steps for Drawing Double Number lines:

1. Draw two straight parallel lines and label with units and arrow as the end.
2. Draw a tick mark through both lines and label with given ratio.
3. Start with 0 and 0 on each number line.
4. Label the units of each number line.
5. Use skip counting to find equivalent ratios and plot them equidistant from each other.  Draw tick marks and label. 
6. Stop when the unknown term has been reached.  Circle and answer with units.

I then guide students through completing the next example in this set.

The student sample, from the Independent Practice problem set, shows what student-created double number lines should look like.

Students work in pairs for the Partner Practice.  As students work, I circulate around the room.  I am looking for:

• Are students correctly labeling the double number lines?
• Are students correctly drawing the double number line? Equal spacing, numbers that correspond to A are along the same line and B along another line?
• Are students correctly skip counting to solve the problem.
• Are students showing clear, logical work?
• Are students providing an answer to the specific question?
• Are students checking for the reasonableness of their answer?

I'm asking students:

• What does the ratio mean in this problem?
• What is the value of each part?  How do you know?
• What is the question asking you to find?
• How did you know to use a double number line?
• How did you find the values for the intervals on the double number line?
• How do you know when to stop skip counting?

After 10 minutes of partner work time, the class comes back together.  I pull a popsicle stick to cold call on a student.  I display one of the double number lines that this student has completed.  I cold call another student and ask if the work shown contains a double number line that's constructed correctly.  I cold call a third student and ask how the original student knew when to stop the double number line.

We discuss problem 2 from this set after work time.  I want students using double number lines when the units are different.  For problem 2, both parts are measured in rows.  Students can use a tape diagram (or ratio table) to solve.  It's very likely that students will have used a double number line to represent this problem.  Numerically, they will come to the right answer.  The conversation after work time will reinforce for students that they should use a double number line for part to part relationships with different units.  Otherwise, we'd be able to model the quantities on one number line, if they have the same unit. 

I then guide students to discuss problem number 4, which involves a total.  We discuss the model that would work best here.  There will be students who used a double number line, labeling tulips and daisies for each respective number line.  They'll have run into the problem of not having a total in the model to work with. 

Students then independently complete the check for understanding at the end of this section.  After 3 minutes of work time, I have students clap out their answer choice for this problem.  

Students work on the Independent Practice problem set.  

After 15 minutes of work time, I have students turn and talk with their partners about their responses from problem number 4 in this set.  I want students to talk about multiple ways to make sense of this problem. 

We then discuss problem 7 as a class.  I want students to articulate how the differences in the pizza problems led students to model the problems in different ways. 

I ask students to work with their partners to summarize when a double number line would be the most appropriate visual model.  I ask 2-3 pairs to share out their thoughts, and we synthesize the responses into one common understanding:  double number lines are used for part to part ratio problems with different units. 

Students work on the Exit Ticket independently to end the class. An Exit Ticket Sample has been included. 

To give students more practice with this model, I send home double number line homework.